factorize the expression x^2+7*x+12

Let's factorize in

ℝ

x^{2} + 7 x + 12

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 7

and

c = 12

Δ = 7^{2} - 4 \cdot 1 \cdot 12

ⓘ Powers are computed before multiplications, additions and subtractions. A power with a positive integer exponent is a repeated product:

7^{2}

7 \cdot 7

49

Δ = 49 - 4 \cdot 12

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 12

4 \cdot (10 + 2)

4 \cdot 10 + 4 \cdot 2

40 + 8

48

Δ = 49 - 48

ⓘ Start from

49

and move back

48

on the number line:

49 - 48

1

Δ = 1

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 7 - \sqrt{1}}{2 \cdot 1}

x_{1} = \frac{- 7 - 1}{2}

ⓘ The numerator and the denominator are computed before the division:

- 7 - 1

-8

x_{1} = - \frac{8}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{8}{2}

-4

x_{1} = -4

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 7 + \sqrt{1}}{2 \cdot 1}

x_{2} = \frac{- 7 + 1}{2}

ⓘ The numerator and the denominator are computed before the division:

- 7 + 1

-6

x_{2} = - \frac{6}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{6}{2}

-3

x_{2} = -3

▶Factors identified:

(x + 3) \cdot (x + 4)